Synthetic aperture radar (SAR) uses relative motion to produce fine
resolution images from microwave frequencies and is a useful tool for
regular monitoring and mapping applications. Unfortunately, if target
distance is estimated poorly, then phase errors are incurred in the data,
producing a blurry reconstruction of the image. In this thesis, we introduce
a new multistatic methodology for determining these phase errors from
interferometry-inspired combinations of signals. To motivate this, we first
consider a more general problem called phase retrieval, in which a signal is
reconstructed from linear measurements whose phases are either unreliable or
unavailable. We make significant theoretical progress on the phase retrieval
problem, to include characterizing injectivity in the complex case, devising
the theory of almost injectivity, and performing a stability analysis. We
then apply certain ideas from phase retrieval to resolve phase errors in
SAR. Specifically, we use bistatic techniques to measure relative phases,
and then we apply a graph-theoretic phase retrieval algorithm to recover the
phase errors. We conclude by devising an image reconstruction procedure
based on this algorithm, and we provide simulations that demonstrate
stability to noise.

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